EWS in measles data
2019-01-22
I use the spaero::get_stats() function to calculate EWS over a moving window for the measles case incidence data from four cities in Niger. I use a bandwidth of 35 weeks, resulting in a 70 week window. All EWS are calculated with backward_only = TRUE so that EWS are only calculated based on data in the past.
# Load packages -----------------------------------------------------------
library(tidyverse)
library(spaero)
# Load data ---------------------------------------------------------------
fname <- "../../data/clean-data/weekly-measles-incidence-niger-cities-clean.RDS"
measles_data <- readRDS(fname) %>%
filter(year > 1994) # drop first NA year, only used for modeling
# Calculate EWS -----------------------------------------------------------
all_stats <- tibble() # empty tibble for storage
for(do_region in unique(measles_data$region)){
cases <- measles_data %>%
filter(region == do_region) %>%
pull(cases)
city_stats <- spaero::get_stats(
x = cases,
center_trend = "local_constant",
center_kernel = "uniform",
center_bandwidth = 35,
stat_trend = "local_constant",
stat_kernel = "uniform",
stat_bandwidth = 35,
lag = 1,
backward_only = TRUE
)$stats
city_stats_tb <- as_tibble(city_stats) %>%
mutate(
time_iter = 1:n(),
date = unique(measles_data$date)
) %>%
gather(key = ews, value = value, -time_iter, -date) %>%
mutate(region = do_region)
all_stats <- bind_rows(all_stats, city_stats_tb)
}Define a function to plot the EWS and the observations.
plot_it <- function(dates, cases, ews, lab, title){
par(mar = c(5,5,2,5))
plot(dates, cases, type = "l", main = title)
par(new = T)
plot(dates, ews, type = "l", col = "red", axes=F, xlab=NA, ylab=NA)
axis(side = 4)
mtext(side = 4, line = 3, lab, cex = 0.7)
}Agadez
do_region <- "Agadez (City)"
dates <- unique(measles_data$date)
par(mfrow = c(4,3))
for(do_ews in unique(all_stats$ews)){
cases <- filter(measles_data, region == do_region) %>% pull(cases)
ews <- filter(all_stats, region == do_region & ews == do_ews) %>% pull(value)
plot_it(dates, cases, ews, lab = "ews", title = do_ews)
}
Maradi
do_region <- "Maradi (City)"
dates <- unique(measles_data$date)
par(mfrow = c(4,3))
for(do_ews in unique(all_stats$ews)){
cases <- filter(measles_data, region == do_region) %>% pull(cases)
ews <- filter(all_stats, region == do_region & ews == do_ews) %>% pull(value)
plot_it(dates, cases, ews, lab = "ews", title = do_ews)
}
Niamey
do_region <- "Niamey (City)"
dates <- unique(measles_data$date)
par(mfrow = c(4,3))
for(do_ews in unique(all_stats$ews)){
cases <- filter(measles_data, region == do_region) %>% pull(cases)
ews <- filter(all_stats, region == do_region & ews == do_ews) %>% pull(value)
plot_it(dates, cases, ews, lab = "ews", title = do_ews)
}
Zinder
do_region <- "Zinder (City)"
dates <- unique(measles_data$date)
par(mfrow = c(4,3))
for(do_ews in unique(all_stats$ews)){
cases <- filter(measles_data, region == do_region) %>% pull(cases)
ews <- filter(all_stats, region == do_region & ews == do_ews) %>% pull(value)
plot_it(dates, cases, ews, lab = "ews", title = do_ews)
}
Post-outbreak EWS
As the plots above show, most EWS get swamped out by the “memory” of outbreaks. To try and get around this, I am going to look at EWS after resetting them following outbreaks. I don’t think this is too contrived because real-world early detection systems would have to do the same thing.
How to ID outbreaks
Following Toby’s suggestion, I am going to look at the distribution of outbreak sizes (i.e., cumulative cases in each year). I can then fit a mixture distribution of two weighted exponentials to the data, with one exponential explaining the large outbreaks and one the small outbreaks. This provides an objective way to define outbreaks for EWS detection.
pop_data <- tibble(
region = c("Agadez (City)", "Maradi (City)", "Niamey (City)", "Zinder (City)"),
pop = c(118224, 267249, 1027000, 322935)
)
cusum_data <- measles_data %>%
group_by(region, year) %>%
summarise(cases = sum(cases)) %>%
left_join(pop_data)Joining, by = "region"ggplot(cusum_data, aes(x = cases/pop)) +
geom_histogram(aes(fill = region))
cusum_cases <- sort(cusum_data$cases / cusum_data$pop)
fit1 <- MASS::fitdistr(cusum_cases, "exponential")
x <- cusum_cases
mixexplik <- function(p,lambda1,lambda2) {
z <- p*dexp(x,lambda1) + (1-p)*dexp(x,lambda2)
return(-sum(log(z)))
}
mle_fit <- suppressWarnings( # ignore warnings about bad guesses
stats4::mle(minuslogl = mixexplik,
start= list(p = 0.2,
lambda1 = as.numeric(fit1$estimate),
lambda2 = as.numeric(fit1$estimate)),
method="Nelder-Mead")
)
get_prob <- function(x, rate1, rate2, p, x_scale, bin_width = 1){
p1 <- pexp((x+bin_width)/x_scale, rate1) - pexp((x-bin_width)/x_scale, rate1)
p2 <- pexp((x+bin_width)/x_scale, rate2) - pexp((x-bin_width)/x_scale, rate2)
mixp <- p*p1+(1-p)*p2
return(c(p1,p2,mixp))
}
x_scale <- 1e05
xstar <- round(x*x_scale)
probs <- sapply(xstar, FUN = get_prob, rate1 = mle_fit@coef["lambda1"],
rate2 = mle_fit@coef["lambda2"], p = mle_fit@coef["p"],
x_scale = x_scale, bin_width = 1)
bound <- qexp(p = 0.5, rate = mle_fit@coef["p"]*mle_fit@coef["lambda1"])
probs_df <- t(log10(probs)) %>%
as_tibble() %>%
dplyr::rename(
"log10prob_large" = V2,
"log10prob_small" = V1,
"log10prob_mix" = p
) %>%
dplyr::mutate(
incidence = x,
outbreak_size = ifelse(incidence < bound, "small", "large"),
plot_prob = ifelse(outbreak_size == "small", log10prob_small, log10prob_large)
)
ggplot(probs_df, aes(x = incidence)) +
geom_line(aes(y = log10prob_mix, color = "forestgreen")) +
geom_point(aes(y = plot_prob, fill = outbreak_size), size = 3, shape = 21, color = "white") +
geom_vline(aes(xintercept = bound), linetype = 2, color = "grey35") +
labs(x = "Case incidence (reports/person)",
y = expression(paste(Log[10]," probability"))) +
scale_fill_discrete(name = NULL, labels = c("Large outbreaks", "Small outbreaks")) +
scale_color_manual(name = NULL, labels = "GMM fit", values = "forestgreen") +
theme_minimal(base_size = 12)
OK, so now I can identify outbreak years in the untransformed (just cases) data for each city.
outbreak_ids <- cusum_data %>%
dplyr::mutate(
incidence = cases/pop,
outbreak_year = ifelse(incidence < bound, FALSE, TRUE)
) %>%
dplyr::select(region, year, outbreak_year)
measles_data <- measles_data %>%
left_join(outbreak_ids)Joining, by = c("region", "year")ggplot(measles_data, aes(x = date, y = cases, color = outbreak_year, group = year)) +
geom_line() +
facet_wrap(~region, scales = "free")
Let’s look at a test case of Niamey post 1995 leading up to the outbreak in 1999.
niamey_test <- measles_data %>%
filter(region == "Niamey (City)") %>%
filter(year > 1998 & year < 2000)
cases <- niamey_test$cases
variance <- spaero::get_stats(
x = cases,
center_trend = "local_constant",
center_kernel = "uniform",
center_bandwidth = 16,
stat_trend = "local_constant",
stat_kernel = "uniform",
stat_bandwidth = 16,
lag = 1,
backward_only = TRUE
)$stats$variance
plot_it(dates = niamey_test$date, cases = cases, ews = log(variance), lab = "log(variance)", title = NULL)
niamey_test <- measles_data %>%
filter(region == "Niamey (City)")
out_vars <- tibble()
for(do_year in unique(niamey_test$year)){
tmp_cases <- niamey_test %>%
filter(year == do_year) %>%
pull(cases)
tmp_stats <- spaero::get_stats(
x = tmp_cases,
center_trend = "local_constant",
center_kernel = "uniform",
center_bandwidth = 16,
stat_trend = "local_constant",
stat_kernel = "uniform",
stat_bandwidth = 16,
lag = 1,
backward_only = TRUE
)$stats
tmp_var <- tibble(
year = do_year,
date = filter(niamey_test, year == do_year)$date,
variance = tmp_stats$variance,
outbreak_year = filter(niamey_test, year == do_year)$outbreak_year
)
out_vars <- bind_rows(out_vars, tmp_var)
}
max_case_week <- niamey_test %>%
group_by(year) %>%
filter(cases == max(cases)) %>%
filter(week_of_year == min(week_of_year))
ggplot() +
geom_line(data = out_vars, aes(x = lubridate::week(date), y = log(variance),
group = year, color = outbreak_year)) +
geom_vline(data = max_case_week, aes(xintercept = week_of_year, color = outbreak_year), linetype = 2)
Hmmm, this seems promising. Perhaps I can fit a GAM through the trajectories and use model selection to determine if an interaction model (outbreak year*time interaction effect) fits better than a GAM with no interaction. Then can use Gavin’s methods for determining time points that are different. Model-based minimum time of difference would be the on-average indicator time, which could be compared to max cases within the year. Should do this with data (like above) and the simulations.
gam_data <- out_vars %>%
dplyr::select(date, variance, outbreak_year) %>%
dplyr::mutate(
week_of_year = lubridate::week(date),
log_variance = log(variance+0.00001),
outbreak_year = as.factor(outbreak_year)
) %>%
dplyr::select(-date)
test_formula <- as.formula(
log_variance ~ outbreak_year + s(week_of_year, by = outbreak_year)
)
gam_model <- mgcv::gam(formula = test_formula, data = gam_data)
pred_df <- tibble(
week_of_year = rep(1:52, times = 2),
outbreak_year = rep(as.factor(c(TRUE, FALSE)), each = 52)
)
preds <- predict(gam_model, newdata = pred_df, type = "response", se.fit = TRUE)
ests <- as.numeric(preds[[1]])
ses <- as.numeric(preds[[2]])
pred_df <- pred_df %>%
mutate(preds = ests,
se = ses)
ggplot(pred_df, aes(x = week_of_year, y = preds, color = outbreak_year)) +
geom_line() +
geom_line(aes(y = preds+(se*2)), linetype = 2) +
geom_line(aes(y = preds-(se*2)), linetype = 2)
But first, look at just case trajectories
ggplot(niamey_test, aes(x = week_of_year, y = cases, color = outbreak_year, group = year)) +
geom_line()
gam_data <- niamey_test %>%
dplyr::select(date, cases, outbreak_year) %>%
dplyr::mutate(
week_of_year = lubridate::week(date),
log_cases = log(cases+0.0001),
outbreak_year = as.factor(outbreak_year)
) %>%
dplyr::select(-date)
test_formula <- as.formula(
log_cases ~ outbreak_year + s(week_of_year, by = outbreak_year)
)
gam_model <- mgcv::gam(formula = test_formula, data = gam_data)
pred_df <- tibble(
week_of_year = rep(1:52, times = 2),
outbreak_year = rep(as.factor(c(TRUE, FALSE)), each = 52)
)
preds <- predict(gam_model, newdata = pred_df, type = "response", se.fit = TRUE)
ests <- as.numeric(preds[[1]])
ses <- as.numeric(preds[[2]])
pred_df <- pred_df %>%
mutate(preds = ests,
se = ses)
ggplot(pred_df, aes(x = week_of_year, y = preds, color = outbreak_year)) +
geom_line() +
geom_line(aes(y = preds+(se*2)), linetype = 2) +
geom_line(aes(y = preds-(se*2)), linetype = 2)
Extra
Maybe there is a variance value that serves as an indicator? Look at high AUCs from the simulations to see what the variances are.
ews <- read_csv("../../results/ews-emergence.csv") %>%
filter(city == "Maradi" & metric == "variance" & half == "second")Parsed with column specification:
cols(
metric = col_character(),
sim = col_integer(),
value = col_double(),
half = col_character(),
city = col_character(),
susc_discount = col_double()
)
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|================================================================================================================================| 100% 13 MBggplot(ews, aes(x = value)) +
geom_histogram() +
facet_wrap(~susc_discount, scales = "free")
med_var <- ews %>%
group_by(susc_discount) %>%
summarise(value = median(value))---
title: "EWS in measles data"
output: html_notebook
date: 2019-01-22
---

I use the `spaero::get_stats()` function to calculate EWS over a moving window for the measles case incidence data from four cities in Niger.
I use a bandwidth of 35 weeks, resulting in a `r 35*2` week window.
All EWS are calculated with `backward_only = TRUE` so that EWS are only calculated based on data in the past.

```{r calc-ews}
# Load packages -----------------------------------------------------------

library(tidyverse)
library(spaero)


# Load data ---------------------------------------------------------------

fname <- "../../data/clean-data/weekly-measles-incidence-niger-cities-clean.RDS"
measles_data <- readRDS(fname) %>%
  filter(year > 1994)  # drop first NA year, only used for modeling


# Calculate EWS -----------------------------------------------------------

all_stats <- tibble()  # empty tibble for storage

for(do_region in unique(measles_data$region)){
  
  cases <- measles_data %>%
    filter(region == do_region) %>%
    pull(cases)
  
  city_stats <- spaero::get_stats(
    x = cases,
    center_trend = "local_constant", 
    center_kernel = "uniform", 
    center_bandwidth = 35, 
    stat_trend = "local_constant", 
    stat_kernel = "uniform", 
    stat_bandwidth = 35, 
    lag = 1, 
    backward_only = TRUE
  )$stats
  
  city_stats_tb <- as_tibble(city_stats) %>%
    mutate(
      time_iter = 1:n(),
      date = unique(measles_data$date)
    ) %>%
    gather(key = ews, value = value, -time_iter, -date) %>%
    mutate(region = do_region)
  
  all_stats <- bind_rows(all_stats, city_stats_tb)
}

```

Define a function to plot the EWS and the observations.

```{r plot-ews-func}
plot_it <- function(dates, cases, ews, lab, title){
  par(mar = c(5,5,2,5))
  plot(dates, cases, type = "l", main = title)
  par(new = T)
  plot(dates, ews, type = "l", col = "red", axes=F, xlab=NA, ylab=NA)
  axis(side = 4)
  mtext(side = 4, line = 3, lab, cex = 0.7)
}
```

### Agadez

```{r agadez, fig.height=8}
do_region <- "Agadez (City)"
dates <- unique(measles_data$date) 
par(mfrow = c(4,3))
for(do_ews in unique(all_stats$ews)){
  cases <- filter(measles_data, region == do_region) %>% pull(cases)
  ews <- filter(all_stats, region == do_region & ews == do_ews) %>% pull(value)
  plot_it(dates, cases, ews, lab = "ews", title = do_ews)
}
```

### Maradi
```{r maradi, fig.height=8}
do_region <- "Maradi (City)"
dates <- unique(measles_data$date) 
par(mfrow = c(4,3))
for(do_ews in unique(all_stats$ews)){
  cases <- filter(measles_data, region == do_region) %>% pull(cases)
  ews <- filter(all_stats, region == do_region & ews == do_ews) %>% pull(value)
  plot_it(dates, cases, ews, lab = "ews", title = do_ews)
}
```

### Niamey
```{r niamey, fig.height=8}
do_region <- "Niamey (City)"
dates <- unique(measles_data$date) 
par(mfrow = c(4,3))
for(do_ews in unique(all_stats$ews)){
  cases <- filter(measles_data, region == do_region) %>% pull(cases)
  ews <- filter(all_stats, region == do_region & ews == do_ews) %>% pull(value)
  plot_it(dates, cases, ews, lab = "ews", title = do_ews)
}
```

### Zinder
```{r zinder, fig.height=8}
do_region <- "Zinder (City)"
dates <- unique(measles_data$date) 
par(mfrow = c(4,3))
for(do_ews in unique(all_stats$ews)){
  cases <- filter(measles_data, region == do_region) %>% pull(cases)
  ews <- filter(all_stats, region == do_region & ews == do_ews) %>% pull(value)
  plot_it(dates, cases, ews, lab = "ews", title = do_ews)
}
```

# Post-outbreak EWS

As the plots above show, most EWS get swamped out by the "memory" of outbreaks.
To try and get around this, I am going to look at EWS after resetting them following outbreaks.
I don't think this is too contrived because real-world early detection systems would have to do the same thing.

## How to ID outbreaks
Following Toby's suggestion, I am going to look at the distribution of outbreak sizes (i.e., cumulative cases in each year).
I can then fit a mixture distribution of two weighted exponentials to the data, with one exponential explaining the large outbreaks and one the small outbreaks.
This provides an objective way to define outbreaks for EWS detection.

```{r}
pop_data <- tibble(
  region = c("Agadez (City)", "Maradi (City)", "Niamey (City)", "Zinder (City)"),
  pop = c(118224, 267249, 1027000, 322935)
)

cusum_data <- measles_data %>%
  group_by(region, year) %>%
  summarise(cases = sum(cases)) %>%
  left_join(pop_data)

ggplot(cusum_data, aes(x = cases/pop)) +
  geom_histogram(aes(fill = region))

cusum_cases <- sort(cusum_data$cases / cusum_data$pop)
fit1 <- MASS::fitdistr(cusum_cases, "exponential") 

x <- cusum_cases
mixexplik <- function(p,lambda1,lambda2) {
  z <- p*dexp(x,lambda1) + (1-p)*dexp(x,lambda2)
  return(-sum(log(z)))
}

mle_fit <- suppressWarnings(  # ignore warnings about bad guesses
  stats4::mle(minuslogl = mixexplik, 
              start= list(p = 0.2,
                          lambda1 = as.numeric(fit1$estimate), 
                          lambda2 = as.numeric(fit1$estimate)), 
              method="Nelder-Mead")
)

get_prob <- function(x, rate1, rate2, p, x_scale, bin_width = 1){
  p1 <- pexp((x+bin_width)/x_scale, rate1) - pexp((x-bin_width)/x_scale, rate1)
  p2 <- pexp((x+bin_width)/x_scale, rate2) - pexp((x-bin_width)/x_scale, rate2)
  mixp <- p*p1+(1-p)*p2
  return(c(p1,p2,mixp))
}

x_scale <- 1e05
xstar <- round(x*x_scale)
probs <- sapply(xstar, FUN = get_prob, rate1 = mle_fit@coef["lambda1"], 
                rate2 = mle_fit@coef["lambda2"], p = mle_fit@coef["p"],
                x_scale = x_scale, bin_width = 1)

bound <- qexp(p = 0.5, rate = mle_fit@coef["p"]*mle_fit@coef["lambda1"])

probs_df <- t(log10(probs)) %>%
  as_tibble() %>%
  dplyr::rename(
    "log10prob_large" = V2,
    "log10prob_small" = V1,
    "log10prob_mix" = p
  ) %>%
  dplyr::mutate(
    incidence = x,
    outbreak_size = ifelse(incidence < bound, "small", "large"),
    plot_prob = ifelse(outbreak_size == "small", log10prob_small, log10prob_large)
  )

ggplot(probs_df, aes(x = incidence)) +
  geom_line(aes(y = log10prob_mix, color = "forestgreen")) +
  geom_point(aes(y = plot_prob, fill = outbreak_size), size = 3, shape = 21, color = "white") +
  geom_vline(aes(xintercept = bound), linetype = 2, color = "grey35") +
  labs(x = "Case incidence (reports/person)", 
       y = expression(paste(Log[10]," probability"))) +
  scale_fill_discrete(name = NULL, labels = c("Large outbreaks", "Small outbreaks")) +
  scale_color_manual(name = NULL, labels = "GMM fit", values = "forestgreen") +
  theme_minimal(base_size = 12)

```

OK, so now I can identify outbreak years in the untransformed (just cases) data for each city.

```{r}
outbreak_ids <- cusum_data %>%
  dplyr::mutate(
    incidence = cases/pop,
    outbreak_year = ifelse(incidence < bound, FALSE, TRUE)
  ) %>%
  dplyr::select(region, year, outbreak_year)

measles_data <- measles_data %>%
  left_join(outbreak_ids)

ggplot(measles_data, aes(x = date, y = cases, color = outbreak_year, group = year)) +
  geom_line() +
  facet_wrap(~region, scales = "free")
```

Let's look at a test case of Niamey post 1995 leading up to the outbreak in 1999.

```{r}
niamey_test <- measles_data %>%
  filter(region == "Niamey (City)") %>%
  filter(year > 1998 & year < 2000)

cases <- niamey_test$cases
variance <- spaero::get_stats(
    x = cases,
    center_trend = "local_constant", 
    center_kernel = "uniform", 
    center_bandwidth = 16, 
    stat_trend = "local_constant", 
    stat_kernel = "uniform", 
    stat_bandwidth = 16, 
    lag = 1, 
    backward_only = TRUE
  )$stats$variance

plot_it(dates = niamey_test$date, cases = cases, ews = log(variance), lab = "log(variance)", title = NULL)

```

```{r}
niamey_test <- measles_data %>%
  filter(region == "Niamey (City)") 

out_vars <- tibble()

for(do_year in unique(niamey_test$year)){
  tmp_cases <- niamey_test %>%
    filter(year == do_year) %>%
    pull(cases)
  
  tmp_stats <- spaero::get_stats(
    x = tmp_cases,
    center_trend = "local_constant", 
    center_kernel = "uniform", 
    center_bandwidth = 16, 
    stat_trend = "local_constant", 
    stat_kernel = "uniform", 
    stat_bandwidth = 16, 
    lag = 1, 
    backward_only = TRUE
  )$stats
  
  tmp_var <- tibble(
    year = do_year,
    date = filter(niamey_test, year == do_year)$date,
    variance = tmp_stats$variance,
    outbreak_year = filter(niamey_test, year == do_year)$outbreak_year
  )
  
  out_vars <- bind_rows(out_vars, tmp_var)
}

max_case_week <- niamey_test %>%
  group_by(year) %>%
  filter(cases == max(cases)) %>%
  filter(week_of_year == min(week_of_year))

ggplot() +
  geom_line(data = out_vars, aes(x = lubridate::week(date), y = log(variance), 
                                 group = year, color = outbreak_year)) +
  geom_vline(data = max_case_week, aes(xintercept = week_of_year, color = outbreak_year), linetype = 2)
```

Hmmm, this seems promising.
Perhaps I can fit a GAM through the trajectories and use model selection to determine if an interaction model (outbreak year*time interaction effect) fits better than a GAM with no interaction.
Then can use Gavin's methods for determining time points that are different.
Model-based minimum time of difference would be the on-average indicator time, which could be compared to max cases within the year.
Should do this with data (like above) and the simulations.

```{r}
gam_data <- out_vars %>%
  dplyr::select(date, variance, outbreak_year) %>%
  dplyr::mutate(
    week_of_year = lubridate::week(date),
    log_variance = log(variance+0.00001),
    outbreak_year = as.factor(outbreak_year)
  ) %>%
  dplyr::select(-date)
  
test_formula <- as.formula(
  log_variance ~ outbreak_year + s(week_of_year, by = outbreak_year)
)
gam_model <- mgcv::gam(formula = test_formula, data = gam_data)

pred_df <- tibble(
  week_of_year = rep(1:52, times = 2),
  outbreak_year = rep(as.factor(c(TRUE, FALSE)), each = 52)
)

preds <- predict(gam_model, newdata = pred_df, type = "response", se.fit = TRUE)
ests <- as.numeric(preds[[1]])
ses <- as.numeric(preds[[2]])
pred_df <- pred_df %>%
  mutate(preds = ests,
         se = ses)

ggplot(pred_df, aes(x = week_of_year, y = preds, color = outbreak_year)) +
  geom_line() +
  geom_line(aes(y = preds+(se*2)), linetype = 2) +
  geom_line(aes(y = preds-(se*2)), linetype = 2)
```

# But first, look at just case trajectories

```{r}
ggplot(niamey_test, aes(x = week_of_year, y = cases, color = outbreak_year, group = year)) +
  geom_line()

gam_data <- niamey_test %>%
  dplyr::select(date, cases, outbreak_year) %>%
  dplyr::mutate(
    week_of_year = lubridate::week(date),
    log_cases = log(cases+0.0001),
    outbreak_year = as.factor(outbreak_year)
  ) %>%
  dplyr::select(-date)
  
test_formula <- as.formula(
  log_cases ~ outbreak_year + s(week_of_year, by = outbreak_year)
)
gam_model <- mgcv::gam(formula = test_formula, data = gam_data)

pred_df <- tibble(
  week_of_year = rep(1:52, times = 2),
  outbreak_year = rep(as.factor(c(TRUE, FALSE)), each = 52)
)

preds <- predict(gam_model, newdata = pred_df, type = "response", se.fit = TRUE)
ests <- as.numeric(preds[[1]])
ses <- as.numeric(preds[[2]])
pred_df <- pred_df %>%
  mutate(preds = ests,
         se = ses)

ggplot(pred_df, aes(x = week_of_year, y = preds, color = outbreak_year)) +
  geom_line() +
  geom_line(aes(y = preds+(se*2)), linetype = 2) +
  geom_line(aes(y = preds-(se*2)), linetype = 2)
```



# Extra
Maybe there is a variance value that serves as an indicator?
Look at high AUCs from the simulations to see what the variances are.

```{r}
ews <- read_csv("../../results/ews-emergence.csv") %>%
  filter(city == "Maradi" & metric == "variance" & half == "second")

ggplot(ews, aes(x = value)) +
  geom_histogram() +
  facet_wrap(~susc_discount, scales = "free")

med_var <- ews %>%
  group_by(susc_discount) %>%
  summarise(value = median(value))
```